Determinants

Every square matrix has a single number attached to it, its determinant. That one number answers the most important question you can ask about a system of equations: does it have a unique solution? It also measures how a transformation scales areas and volumes, gives formulas for cross products and triple products, and turns up later in multivariable calculus (the Jacobian), in differential equations (the Wronskian) and in Lesson 5, where it is the key to finding eigenvalues.

This chapter defines the determinant, shows how to compute it efficiently by hand, explains its geometric meaning, and collects the properties that make it useful. Along the way you will compute the area of a surveyed lot and the volume of a box from its edge vectors.

The 2×2 determinant

For a 2×2 matrix the determinant is the product of the main diagonal minus the product of the other diagonal:

You met this number in the 2×2 inverse formula in Lesson 3, where it appeared in the denominator. That is no accident: the inverse exists exactly when \(ad - bc \ne 0\).

What it measures

Place the two columns \((a, c)\) and \((b, d)\) tail to tail. They span a parallelogram, and the absolute value of the determinant is its area. The sign records orientation: positive if turning from the first column to the second is counterclockwise, negative if clockwise.

Since the matrix sends the unit square (area 1) to this parallelogram, the determinant is the factor by which the transformation scales every area. A shape of area 3 becomes a shape of area \(3|\det A|\). A rotation has determinant 1 (areas are preserved); the stretch \(\begin{pmatrix} 2 & 0 \\ 0 & 1 \end{pmatrix}\) has determinant 2; a reflection has determinant \(-1\) (area preserved, orientation flipped).

If the determinant is zero, the columns are parallel and the parallelogram is flat. The transformation squashes the whole plane onto a line, and that is why it cannot be undone: many different vectors land on the same point, so there is no inverse.

The 3×3 determinant by cofactor expansion

For a 3×3 matrix, expand along the first row. Each entry multiplies the 2×2 determinant left when you cross out its row and column (its minor), with signs alternating \(+, -, +\):

A minor with its sign attached is called a cofactor. You can expand along any row or any column, not just the first, as long as you use the checkerboard of signs:

Choose the row or column with the most zeros, since each zero kills a whole term.

For a 3×3 matrix the absolute value of the determinant is the volume of the box (parallelepiped) spanned by the three columns, and it is the factor by which the transformation scales volumes. It equals the scalar triple product from Lesson 1: \(\det[\mathbf{a}\;\mathbf{b}\;\mathbf{c}] = \mathbf{a}\cdot(\mathbf{b}\times\mathbf{c})\). The cross product formula written as a determinant with \(\mathbf{i}, \mathbf{j}, \mathbf{k}\) in the top row is the same cofactor expansion.

Properties of determinants

Cofactor expansion of an \(n \times n\) matrix needs roughly \(n!\) multiplications. For \(n = 10\) that is over three million, and for \(n = 25\) a computer doing a billion operations per second would need hundreds of millions of years. The properties below lead to a far faster method, and they are useful in their own right.

Row operations

Each of the three row operations from Lesson 2 has a predictable effect:

• Swapping two rows multiplies the determinant by \(-1\).
• Multiplying a row by \(c\) multiplies the determinant by \(c\).
• Adding a multiple of one row to another leaves the determinant unchanged.

Two consequences follow at once. A matrix with a row of zeros, or with two equal or proportional rows, has determinant zero. And multiplying the whole \(n \times n\) matrix by \(c\) scales every row, so \(\det(cA) = c^n\det A\), not \(c\det A\).

Triangular matrices

The determinant of a triangular matrix (upper or lower) is the product of its diagonal entries. Expanding down the first column repeatedly leaves only that product. This includes diagonal matrices and the identity, whose determinant is 1.

Products, transposes and inverses

The product rule matches the geometry: doing transformation \(B\) and then \(A\) scales areas by \(\det B\) and then by \(\det A\). For example, \(\det\begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix} = -2\) and \(\det\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} = -1\), and their product \(\begin{pmatrix} 2 & 1 \\ 4 & 3 \end{pmatrix}\) has determinant \(6 - 4 = 2 = (-2)(-1)\). There is no such rule for sums: \(\det(A + B)\) is generally not \(\det A + \det B\).

Because \(\det(A^T) = \det A\), everything said about rows is also true of columns.

Computing determinants by row reduction

Reduce the matrix to triangular form with row operations, keeping track of swaps and scalings, then multiply the diagonal entries. This needs about \(n^3/3\) operations, the same as elimination, and it is how software computes determinants.

Determinants and invertibility

Everything so far comes together in one theorem. For a square matrix \(A\), the following statements are all equivalent. If one is true, they all are:

• \(\det A \ne 0\).
• \(A\) is invertible.
• \(A\mathbf{x} = \mathbf{b}\) has exactly one solution for every \(\mathbf{b}\).
• \(A\mathbf{x} = \mathbf{0}\) has only the solution \(\mathbf{x} = \mathbf{0}\).
• Elimination gives a pivot in every column (the rank is \(n\)).
• The columns of \(A\) do not lie in a common line, plane or lower-dimensional space.

So the determinant is a one-number test for whether a system is well posed. In design work it is common to look for parameter values that make a determinant zero, because those are the values where something goes wrong: a mechanism locks or becomes free to move, a structure buckles, a control system loses the ability to steer a state.

Cramer's rule

For a system \(A\mathbf{x} = \mathbf{b}\) with \(\det A \ne 0\), each unknown is a ratio of determinants:

where \(A_i\) is \(A\) with its \(i\)-th column replaced by \(\mathbf{b}\).

Cramer's rule is elegant and handy for 2×2 systems, for finding just one unknown, or for getting a formula in terms of symbols (as in circuit transfer functions). For numerical work beyond 3×3 it is far slower than elimination. As a check, for the 3×3 system solved in Lesson 2, \(\det A = 7\) and replacing column 1 by \((6, 3, 2)\) also gives 7, so \(x = 1\), agreeing with elimination.

Application: areas from survey coordinates

Surveyors find the area of a lot from the coordinates of its corners using the shoelace formula, which is a sum of 2×2 determinants. For vertices \((x_1, y_1), \dots, (x_n, y_n)\) listed in order around the boundary:

with \((x_{n+1}, y_{n+1}) = (x_1, y_1)\). Each term is twice the signed area of the triangle from the origin to one edge. Triangles outside the shape are counted once positive and once negative, so they cancel.

The same formula is how CAD programs and GIS software compute polygon areas, and in Calculus III it reappears as Green's theorem.

Looking ahead: the Jacobian

Because the determinant is the local area-scaling factor of a transformation, it appears whenever you change variables in a multiple integral. Switching to polar coordinates, the small rectangle \(dr\,d\theta\) maps to a curved patch of area \(r\,dr\,d\theta\). That factor \(r\) is the determinant of the matrix of partial derivatives, called the Jacobian. You will use it in Calculus III to find masses, centroids and moments of inertia of circular and spherical parts.